Finite Group Databasebeta

← C23×#9532 #19157158

Subgroups — C23×#9532 #19157158

Order 5888.

444 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3114C211yesmaximal, characteristic
2C23×#2967C211yesmaximal, characteristic, complemented
3C23×#1535C211yesmaximal, characteristic, complemented
4C23×#1490C211yesmaximal, characteristic, complemented
5C23×#1466C211yesmaximal, characteristic, complemented
6C23×#1308C211yesmaximal, characteristic, complemented
7C23×#1291C211yesmaximal, characteristic, complemented
8C23×#529C2211yescharacteristic
9C2×C4×D4×C23C2211yescharacteristic
10C23×#483C411yescharacteristic
11C23×#480C411yescharacteristic
12C23×#478—21no—
13C23×#463—21no—
14C23×#450C411yescharacteristic
15C23×#447C411yescharacteristic
16C23×#446—21no—
17C23×#442—21no—
18C23×#440—21no—
19C23×#436—21no—
20C23×#435—21no—
21C23×#435—21no—
22C23×#433—21no—
23C23×#432C2211yescharacteristic
24C23×#432C2211yescharacteristic
25C23×#432C2211yescharacteristic
26C23×#432C2211yescharacteristic
27C23×#405—21no—
28C4×M16×C23C2211yescharacteristic
29C23×#330—21no—
30C23×#329—21no—
31C23×D4⋊C8—21no—
32C23×D4⋊C8—21no—
33C23×#323—21no—
34C23×#323—21no—
35C22×D4×C23C2×C411yescharacteristic
36C23×C4×C23—21no—
37C2×SD16×C23—21no—
38C2×D8×C23—21no—
39C2×M16×C23—21no—
40C2×M16×C23—21no—
41C22×C8×C23—21no—
42C23×#126—21no—
43C23×#125D411yescharacteristic
44C23×#124D411yescharacteristic
45C23×#123—21no—
46C23×C22⋊Q8C2×C411yescharacteristic
47C23×#121C2×C411yescharacteristic
48C4×Q8×C23D411yescharacteristic
49C4×D4×C23—21no—
50C4×D4×C23—21no—
51C4×D4×C23—21no—
52C4×D4×C23—21no—
53C4×D4×C23—21no—
54C4×D4×C23D411yescharacteristic
55C23×#117—21no—
56C2×C4⋊C4×C23C2×C411yescharacteristic
57C2×C22⋊C4×C23—21no—
58C2×C42×C23C2311yescharacteristic
59C23×#107—21no—
60C23×#106—21no—
61C23×C4⋊C8—21no—
62C23×C4⋊C8—21no—
63C23×C4⋊C8—21no—
64C23×C4⋊C8—21no—
65C23×C4⋊C8—21no—
66C23×C4⋊C8—21no—
67C23×#103—21no—
68C23×#102—21no—
69C23×#102—21no—
70C23×#102—21no—
71C23×C22⋊C8—21no—
72C23×C22⋊C8—21no—
73C23×C22⋊C8—21no—
74C23×C22⋊C8—21no—
75C23×C22⋊C8C2×C411yescharacteristic
76C23×C22⋊C8C2×C411yescharacteristic
77C23×C22⋊C8D411yescharacteristic
78C23×C22⋊C8D411yescharacteristic
79C23×#97—21no—
80C23×#97—21no—
81C23×#97—21no—
82C23×#97—21no—
83C23×#97—21no—
84C23×#97—21no—
85C4×C8×C23—21no—
86C4×C8×C23D411yescharacteristic
87C4×C8×C23D411yescharacteristic
88C24×C23—21no—
89C2×Q8×C23C22⋊C411yescharacteristic
90C2×D4×C23—21no—
91C2×D4×C23—21no—
92C2×D4×C23—21no—
93C2×D4×C23—42no—
94C2×D4×C23C22⋊C411yescharacteristic
95C22×C4×C23—21no—
96C22×C4×C23—21no—
97C22×C4×C23—41no—
98C22×C4×C23—41no—
99C22×C4×C23—82no—
100C22×C4×C23C22×C411yescharacteristic
101C22×C4×C23C2×D411yescharacteristic
102C22×C4×C23#4111yescharacteristic
103SD16×C23—42no—
104D8×C23—42no—
105M16×C23—42no—
106M16×C23—42no—
107C2×C8×C23—21no—
108C2×C8×C23—21no—
109C2×C8×C23—21no—
110C2×C8×C23—21no—
111C2×C8×C23—21no—
112C2×C8×C23—21no—
113C2×C8×C23—42no—
114C2×C8×C23—41no—
115C2×C8×C23—41no—
116C4⋊C4×C23—21no—
117C4⋊C4×C23—21no—
118C4⋊C4×C23—21no—
119C4⋊C4×C23—21no—
120C4⋊C4×C23—21no—
121C4⋊C4×C23—21no—
122C4⋊C4×C23C22⋊C411yescharacteristic
123C4⋊C4×C23C22⋊C411yescharacteristic
124C22⋊C4×C23—21no—
125C22⋊C4×C23—21no—
126C22⋊C4×C23—21no—
127C22⋊C4×C23—21no—
128C22⋊C4×C23—21no—
129C22⋊C4×C23—41no—
130C22⋊C4×C23—41no—
131C42×C23—21no—
132C42×C23C2×D411yescharacteristic
133C42×C23C2×D411yescharacteristic
134C42×C23C2×D411yescharacteristic
135C42×C23#4111yescharacteristic
136#9532C2311yesmaximal, characteristic, direct product factor, Sylow
137C23×C23—22no—
138C23×C23—21no—
139C23×C23—41no—
140C23×C23—41no—
141C23×C23—81no—
142C23×C23—81no—
143C23×C23C4×D411yescharacteristic
144Q8×C23—41no—
145D4×C23—22no—
146D4×C23—22no—
147D4×C23—41no—
148D4×C23—41no—
149D4×C23—41no—
150D4×C23—81no—
151C2×C4×C23—21no—
152C2×C4×C23—21no—
153C2×C4×C23—21no—
154C2×C4×C23—41no—
155C2×C4×C23—41no—
156C2×C4×C23—41no—
157C2×C4×C23—41no—
158C2×C4×C23—41no—
159C2×C4×C23—41no—
160C2×C4×C23—82no—
161C2×C4×C23—82no—
162C2×C4×C23—81no—
163C2×C4×C23C2×C22⋊C411yescharacteristic
164C2×C4×C23C4×D411yescharacteristic
165C2×C4×C23#12011yescharacteristic
166C2×C4×C23#12111yescharacteristic
167C2×C4×C23#12111yescharacteristic
168C2×C4×C23#12311yescharacteristic
169C8×C23—22no—
170C8×C23—41no—
171C8×C23—41no—
172C8×C23—41no—
173C8×C23—41no—
174C8×C23—41no—
175C8×C23—81no—
176C8×C23—81no—
177#3114C2×C2311yescharacteristic
178#2967C2×C2311yescharacteristic, complemented
179#1535C2×C2311yescharacteristic, complemented
180#1490C2×C2311yescharacteristic, complemented
181#1466C2×C2311yescharacteristic, complemented
182#1308C2×C2311yescharacteristic, complemented
183#1291C2×C2311yescharacteristic, complemented
184C22×C23—21no—
185C22×C23—21no—
186C22×C23—21no—
187C22×C23—42no—
188C22×C23—41no—
189C22×C23—41no—
190C22×C23—41no—
191C22×C23—82no—
192C22×C23—82no—
193C22×C23—81no—
194C22×C23—81no—
195C22×C23#38611yescharacteristic
196C4×C23—21no—
197C4×C23—21no—
198C4×C23—21no—
199C4×C23—21no—
200C4×C23—21no—
201C4×C23—21no—
202C4×C23—41no—
203C4×C23—41no—
204C4×C23—41no—
205C4×C23—81no—
206#529C22×C2311yescharacteristic
207C2×C4×D4C22×C2311yescharacteristic
208#483C4×C2311yescharacteristic
209#480C4×C2311yescharacteristic
210#478—21no—
211#463—21no—
212#450C4×C2311yescharacteristic
213#447C4×C2311yescharacteristic
214#446—21no—
215#442—21no—
216#440—21no—
217#436—21no—
218#435—21no—
219#435—21no—
220#433—21no—
221#432C22×C2311yescharacteristic
222#432C22×C2311yescharacteristic
223#432C22×C2311yescharacteristic
224#432C22×C2311yescharacteristic
225#405—21no—
226C4×M16C22×C2311yescharacteristic
227#330—21no—
228#329—21no—
229D4⋊C8—21no—
230D4⋊C8—21no—
231#323—21no—
232#323—21no—
233C2×C23—41no—
234C2×C23—81no—
235C2×C23—82no—
236C2×C23#189111yescharacteristic
237C2×C23#190311yescharacteristic
238C2×C23#190611yescharacteristic
239C22×D4C2×C4×C2311yescharacteristic
240C23×C4—21no—
241C2×SD16—21no—
242C2×D8—21no—
243C2×M16—21no—
244C2×M16—21no—
245C22×C8—21no—
246#126—21no—
247#125D4×C2311yescharacteristic
248#124D4×C2311yescharacteristic
249#123—21no—
250C22⋊Q8C2×C4×C2311yescharacteristic
251#121C2×C4×C2311yescharacteristic
252C4×Q8D4×C2311yescharacteristic
253C4×D4—21no—
254C4×D4—21no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4D4×C2311yescharacteristic
259#117—21no—
260C2×C4⋊C4C2×C4×C2311yescharacteristic
261C2×C22⋊C4—21no—
262C2×C42C23×C2311yescharacteristic
263#107—21no—
264#106—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271#103—21no—
272#102—21no—
273#102—21no—
274#102—21no—
275C22⋊C8—21no—
276C22⋊C8—21no—
277C22⋊C8—21no—
278C22⋊C8—21no—
279C22⋊C8C2×C4×C2311yescharacteristic
280C22⋊C8C2×C4×C2311yescharacteristic
281C22⋊C8D4×C2311yescharacteristic
282C22⋊C8D4×C2311yescharacteristic
283#97—21no—
284#97—21no—
285#97—21no—
286#97—21no—
287#97—21no—
288#97—21no—
289C4×C8—21no—
290C4×C8D4×C2311yescharacteristic
291C4×C8D4×C2311yescharacteristic
292C23#953211yescharacteristic, direct product factor, Sylow
293C24—21no—
294C2×Q8C22⋊C4×C2311yescharacteristic
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—42no—
299C2×D4C22⋊C4×C2311yescharacteristic
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—82no—
305C22×C4C22×C4×C2311yescharacteristic
306C22×C4C2×D4×C2311yescharacteristic
307C22×C4#41×C2311yescharacteristic
308SD16—42no—
309D8—42no—
310M16—42no—
311M16—42no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—42no—
319C2×C8—41no—
320C2×C8—41no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4C22⋊C4×C2311yescharacteristic
328C4⋊C4C22⋊C4×C2311yescharacteristic
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C42—21no—
337C42C2×D4×C2311yescharacteristic
338C42C2×D4×C2311yescharacteristic
339C42C2×D4×C2311yescharacteristic
340C42#41×C2311yescharacteristic
341C23—22no—
342C23—21no—
343C23—41no—
344C23—41no—
345C23—81no—
346C23—81no—
347C23C4×D4×C2311yescharacteristic
348Q8—41no—
349D4—22no—
350D4—22no—
351D4—41no—
352D4—41no—
353D4—41no—
354D4—81no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4C2×C22⋊C4×C2311yescharacteristic
368C2×C4C4×D4×C2311yescharacteristic
369C2×C4C23×#12011yescharacteristic
370C2×C4C23×#12111yescharacteristic
371C2×C4C23×#12111yescharacteristic
372C2×C4C23×#12311yescharacteristic
373C8—22no—
374C8—41no—
375C8—41no—
376C8—41no—
377C8—41no—
378C8—41no—
379C8—81no—
380C8—81no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—42no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—82no—
389C22—82no—
390C22—81no—
391C22—81no—
392C22C23×#38611yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—81no—
403C2—41no—
404C2—81no—
405C2—82no—
406C2C23×#189111yescharacteristic
407C2C23×#190311yescharacteristic
408C2C23×#190611yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC23×#447C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C23×#2967C2semidirect
3. C23×#1535C2semidirect
4. C23×#1490C2semidirect
5. C23×#1466C2semidirect
6. C23×#1308C2semidirect
7. C23×#1291C2semidirect
136. #9532C23direct (N × H)
178. #2967C2×C23semidirect
179. #1535C2×C23semidirect
180. #1490C2×C23semidirect
181. #1466C2×C23semidirect
182. #1308C2×C23semidirect
183. #1291C2×C23semidirect
292. C23#9532direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C23×#31142
C23×#29672
C23×#15352
C23×#14902
C23×#14662
C23×#13082
C23×#12912
#953223